Higher Engineering Mathematics, Sixth Edition

The normal distribution 567

30 32 34 36 38 40 42

0.01

0.05

0.1

0.2

0.5

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

99.8

99.9

99.99

Percentage cumulative frequency

Upper class boundary

Q

P

R

Figure 58.6

paper, the standard deviation of the distribution is
given by:
(
variable value for 84% cumulative frequency−
variable value for 16% cumulative frequency

)

2

Problem 5. Use normal probability paper to

determine whether the data given below, which

refers to the masses of 50 copper ingots, is

approximately normally distributed. If the data is

normally distributed, determine the mean and

standard deviation of the data from the graph drawn.

Class mid-point value (kg) Frequency

29.5 2

30.5 4

31.5 6

32.5 8

33.5 9

34.5 8

0.01

0.05

0.1

0.2

0.5

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

99.9

99.99

Upper class boundary

Percentage cumulative frequency

B

A

C

10 20 30 40 50 60 70 80 90 100 110

Figure 58.7

Class mid-point value (kg) Frequency

35.5 6

36.5 4

37.5 2

38.5 1

To test the normality of a distribution, the upper class

boundary/percentage cumulative frequency values are

plotted on normal probability paper. The upper class

boundary values are: 30, 31, 32,..., 38, 39. The corre-

sponding cumulative frequency values (for ‘less than’

the upper class boundary values) are: 2,( 4 + 2 )=6,

( 6 + 4 + 2 )=12, 20, 29, 37, 43, 47, 49 and 50. The cor-

responding percentage cumulative frequency values are

2

50

× 100 =4,

6

50

× 100 =12, 24, 40, 58, 74, 86, 94, 98

and 100%.

The co-ordinates of upper class boundary/percentage

cumulative frequency values are plotted as shown